1. Differentiating Mathematics at the Middle
and High School Levels
Raising Student Achievement Conference
St. Charles, IL
December 4, 2007
"In the end, all learners need your
energy, your heart and your mind.
They have that in common because
they are young humans. How they
need you however, differs. Unless
we understand and respond to
those differences, we fail many
learners." *
* Tomlinson, C.A. (2001). How to differentiate instruction in mixed ability
classrooms (2nd Ed.). Alexandria, VA: ASCD.
Nanci Smith
Educational Consultant
Curriculum and Professional Development
Cave Creek, AZ
nanci_mathmaster@yahoo.com

2. Differentiation of Instruction
Is a teacher’s response to learner’s needs
guided by general principles of differentiation
Respectful tasks Flexible grouping Continual assessment
Teachers Can Differentiate Through:
Content Process Product
According to Students’
Readiness Interest Learning Profile

3. What’s the point of differentiating
in these different ways?
Learning
Readines s Interes t
Profile
Growth Motivation E fficiency

4. Key Principles of a
Differentiated Classroom
• The teacher understands, appreciates,
and builds upon student differences.
Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD

5. READINESS
What does READINESS mean?
It is the student’s entry point
relative to a particular
understanding or skill.
C.A.Tomlinson, 1999

6. A Few Routes to READINESS
DIFFERENTIATION
Varied texts by reading level
Varied supplementary materials
Varied scaffolding
• reading
• writing
• research
• technology
Tiered tasks and procedures
Flexible time use
Small group instruction
Homework options
Tiered or scaffolded assemssment
Compacting
Mentorships
Negotiated criteria for quality
Varied graphic organizers

7. Providing support
needed for a
student to succeed
in work slightly
For example… beyond his/her
•Directions that give more structure – or comfort zone.
less
•Tape recorders to help with reading or writing beyond the student’s grasp
•Icons to help interpret print
•Reteaching / extending teaching
•Modeling
•Clear criteria for success
•Reading buddies (with appropriate directions)
•Double entry journals with appropriate challenge
•Teaching through multiple modes
•Use of manipulatives when needed
•Gearing reading materials to student reading level
•Use of study guides
•Use of organizers
•New American Lecture
Tomlinson, 2000

8. Compacting
1. Identify the learning objectives or standards ALL students
must learn.
2. Offer a pretest opportunity OR plan an alternate path through
the content for those students who can learn the required
material in less time than their age peers.
3. Plan and offer meaningful curriculum extensions for kids who
qualify.
**Depth and Complexity
Applications of the skill being taught
Learning Profile tasks based on understanding the
process instead of skill practice
Differing perspectives, ideas across time, thinking
like a mathematician
**Orbitals and Independent studies.
9. Eliminate all drill, practice, review, or preparation for students
who have already mastered such things.
10. Keep accurate records of students’ compacting activities:
document mastery.
Strategy: Compacting

9. Developing a Tiered Activity
1
Select the activity organizer 2
•concept Think about your students/use assessments
Essential to building
•generalization a framework of skills
understanding • readiness range reading
thinking
• interests information
• learning profile
• talents
3
Create an activity that is
• interesting 4
• high level High skill/
• causes students to use Chart the Complexity
key skill(s) to understand complexity of
a key idea the activity
Low skill/
complexity
5
Clone the activity along the ladder as
needed to ensure challenge and success
for your students, in
• materials – basic to advanced 6
• form of expression – from familiar to
unfamiliar Match task to student based on
• from personal experience to removed
from personal experience student profile and task
•equalizer requirements

10. The Equalizer
1. Foundational Transformational
Information, Ideas, Materials, Applications
3. Concrete Abstract
Representations, Ideas, Applications, Materials
5. Simple Complex
Resources, Research, Issues, Problems, Skills, Goals
7. Single Facet Multiple Facets
Directions, Problems, Application, Solutions, Approaches, Disciplinary Connections
9. Small Leap Great Leap
Application, Insight, Transfer
11. More Structured More Open
Solutions, Decisions, Approaches
13. Less Independence Greater Independence
Planning, Designing, Monitoring
15. Slow Pace of Study, Pace of Thought Quick

11. Adding Fractions
Green Group
Use Cuisinaire rods or fraction
circles to model simple fraction
addition problems. Begin with Blue Group
common denominators and work
up to denominators with common Manipulatives such as Cuisinaire
factors such as 3 and 6. rods and fraction circles will be
available as a resource for the
group. Students use factor trees
Explain the pitfalls and hurrahs of and lists of multiples to find
adding fractions by making a common denominators. Using this
picture book. approach, pairs and triplets of
Red Group fractions are rewritten using
common denominators. End by
Use Venn diagrams to model adding several different problems
LCMs (least common multiple). of increasing challenge and length.
Explain how this process can be
used to find common
denominators. Use the method on Suzie says that adding fractions is
more challenging addition like a game: you just need to know
problems. the rules. Write game instructions
explaining the rules of adding
fractions.
Write a manual on how to add
fractions. It must include why a
common denominator is needed,
and at least three ways to find it.

12. Graphing with a Point and a Slope
All groups:
• Given three equations in slope-intercept form, the
students will graph the lines using a T-chart. Then
they will answer the following questions:
• What is the slope of the line?
• Where is slope found in the equation?
• Where does the line cross the y-axis?
• What is the y-value of the point when x=0? (This
is the y-intercept.)
• Where is the y-value found in the equation?
• Why do you think this form of the equation is
called the “slope-intercept?”

13. Graphing with a Point and a Slope
Struggling Learners: Given the points
• (-2,-3), (1,1), and (3,5), the students will plot the points
and sketch the line. Then they will answer the following
questions:
• What is the slope of the line?
• Where does the line cross the y-axis?
• Write the equation of the line.
The students working on this particular task should repeat this
process given two or three more points and/or a point and a slope.
They will then create an explanation for how to graph a line starting
with the equation and without finding any points using a T-chart.

14. Graphing with a Point and a Slope
Grade-Level Learners: Given an equation of a line in
slope-intercept form (or several equations), the students
in this group will:
• Identify the slope in the equation.
• Identify the y-intercept in the equation.
• Write the y-intercept in coordinate form (0,y) and plot
the point on the y-axis.
• use slope to find two additional points that will be on the
line.
• Sketch the line.
When the students have completed the above tasks, they will
summarize a way to graph a line from an equation without using a
T-chart.

15. Graphing with a Point and a Slope
Advanced Learners: Given the slope-intercept form of the
equation of a line, y=mx+b, the students will answer the
following questions:
• The slope of the line is represented by which variable?
• The y-intercept is the point where the graph crosses the
y-axis. What is the x-coordinate of the y-intercept? Why
will this always be true?
• The y-coordinate of the y-intercept is represented by
which variable in the slope-intercept form?
Next, the students in this group will complete the following
tasks given equations in slope-intercept form:
• Identify the slope and the y-intercept.
• Plot the y-intercept.
• Use the slope to count rise and run in order to find the
second and third points.
• Graph the line.

16. BRAIN RESEARCH SHOWS THAT. . .
Eric Jensen, Teaching With the Brain in Mind, 1998
Choices vs. Required
content, process, product no student voice
groups, resources environment restricted resources
Relevant vs. Irrelevant
meaningful impersonal
connected to learner out of context
deep understanding only to pass a test
Engaging vs. Passive
emotional, energetic low interaction
hands on, learner input lecture seatwork
EQUALS
Increased intrinsic Increased
MOTIVATION APATHY & RESENTMENT

17. -CHOICE-
The Great Motivator!
• Requires children to be aware of their own readiness, interests, and
learning profiles.
• Students have choices provided by the teacher. (YOU are still in
charge of crafting challenging opportunities for all kiddos – NO
taking the easy way out!)
• Use choice across the curriculum: writing topics, content writing
prompts, self-selected reading, contract menus, math problems,
spelling words, product and assessment options, seating, group
arrangement, ETC . . .
• GUARANTEES BUY-IN AND ENTHUSIASM FOR LEARNING!
• Research currently suggests that CHOICE should be offered 35%
of the time!!

18. Assessments
The assessments used in this learning profile
section can be downloaded at:
www.e2c2.com/fileupload.asp
Download the file entitled “Profile
Assessments for Cards.”

19. How Do You Like to Learn?
1. I study best when it is quiet. Yes No
2. I am able to ignore the noise of
other people talking while I am working. Yes No
3. I like to work at a table or desk. Yes No
4. I like to work on the floor. Yes No
5. I work hard by myself. Yes No
6. I work hard for my parents or teacher. Yes No
7. I will work on an assignment until it is completed, no
matter what. Yes No
8. Sometimes I get frustrated with my work
and do not finish it. Yes No
9. When my teacher gives an assignment, I like to
have exact steps on how to complete it. Yes No
10. When my teacher gives an assignment, I like to
create my own steps on how to complete it. Yes No
11. I like to work by myself. Yes No
12. I like to work in pairs or in groups. Yes No
13. I like to have unlimited amount of time to work on
an assignment. Yes No
14. I like to have a certain amount of time to work on
an assignment. Yes No
15. I like to learn by moving and doing. Yes No
16. I like to learn while sitting at my desk. Yes No

20. My Way
An expression Style Inventory
K.E. Kettle J.S. Renzull, M.G. Rizza
University of Connecticut
Products provide students and professionals with a way to express what they have
learned to an audience. This survey will help determine the kinds of products
YOU are interested in creating.
My Name is: ____________________________________________________
Instructions:
Read each statement and circle the number that shows to what extent YOU are
interested in creating that type of product. (Do not worry if you are unsure of how
to make the product).
Not At All Interested Of Little Interest Moderately Interested Interested Very Interested
1. Writing Stories 1 2 3 4 5
2. Discussing what I 1 2 3 4 5
have learned
3. Painting a picture 1 2 3 4 5
4. Designing a 1 2 3 4 5
computer software
project
5. Filming & editing a 1 2 3 4 5
video
6. Creating a company 1 2 3 4 5
7. Helping in the 1 2 3 4 5
community
8. Acting in a play 1 2 3 4 5

21. Not At All Interested Of Little Interest Moderately Interested Interested Very Interested
9. Building an 1 2 3 4 5
invention
10. Playing musical 1 2 3 4 5
instrument
11. Writing for a 1 2 3 4 5
newspaper
12. Discussing ideas 1 2 3 4 5
13. Drawing pictures 1 2 3 4 5
for a book
14. Designing an 1 2 3 4 5
interactive computer
project
15. Filming & editing 1 2 3 4 5
a television show
16. Operating a 1 2 3 4 5
business
17. Working to help 1 2 3 4 5
others
18. Acting out an 1 2 3 4 5
event
19. Building a project 1 2 3 4 5
20. Playing in a band 1 2 3 4 5
21. Writing for a 1 2 3 4 5
magazine
22. Talking about my 1 2 3 4 5
project
23. Making a clay 1 2 3 4 5
sculpture of a
character

22. Not At All Interested Of Little Interest Moderately Interested Interested Very Interested
24. Designing 1 2 3 4 5
information for the
computer internet
25. Filming & editing 1 2 3 4 5
a movie
26. Marketing a 1 2 3 4 5
product
27. Helping others by 1 2 3 4 5
supporting a social
cause
28. Acting out a story 1 2 3 4 5
29. Repairing a 1 2 3 4 5
machine
30. Composing music 1 2 3 4 5
31. Writing an essay 1 2 3 4 5
32. Discussing my 1 2 3 4 5
research
33. Painting a mural 1 2 3 4 5
34. Designing a 1 2 3 4 5
computer
35. Recording & 1 2 3 4 5
editing a radio show
36. Marketing an idea 1 2 3 4 5
37. Helping others by 1 2 3 4 5
fundraising
38. Performing a skit 1 2 3 4 5

23. Not At All Interested Of Little Interest Moderately Interested Interested Very Interested
39. Constructing a 1 2 3 4 5
working model.
40. Performing music 1 2 3 4 5
41. Writing a report 1 2 3 4 5
42. Talking about my 1 2 3 4 5
experiences
43. Making a clay 1 2 3 4 5
sculpture of a scene
44. Designing a multi- 1 2 3 4 5
media computer show
45. Selecting slides 1 2 3 4 5
and music for a slide
show
46. Managing 1 2 3 4 5
investments
47. Collecting 1 2 3 4 5
clothing or food to
help others
48. Role-playing a 1 2 3 4 5
character
49. Assembling a kit 1 2 3 4 5
50. Playing in an 1 2 3 4 5
orchestra
Products Total
Instructions: My Written 1. ___ 11. ___ 21. ___ 31. ___ 41. ___ _____
Way …A Profile Oral 2. ___ 12. ___ 22. ___ 32. ___ 42. ___ _____
Artistic 3. ___ 13. ___ 23. ___ 33. ___ 43. ___ _____
Write your score Computer 4. ___ 14. ___ 24. ___ 34. ___ 44. ___ _____
beside each Audio/Visual 5. ___ 15. ___ 25. ___ 35. ___ 45. ___ _____
number. Add each Commercial 6. ___ 16. ___ 26. ___ 36. ___ 46. ___ _____
Row to determine Service 7. ___ 77. ___ 27. ___ 37. ___ 47. ___ _____
your expression Dramatization 8. ___ 18. ___ 28. ___ 38. ___ 48. ___ _____
style profile. Manipulative 9. ___ 19. ___ 29. ___ 39. ___ 49. ___ _____
Musical 10.___ 20. ___ 30 . ___ 40. ___ 50. ___ _____

24. Learner Profile Card
Gender Stripe
Auditory, Visual, Kinesthetic Analytical, Creative, Practical
Modality Sternberg
Student’s
Interests
Multiple Intelligence Preference
Array
Gardner Inventory
Nanci Smith,Scottsdale,AZ

25. Differentiation Using
LEARNING PROFILE
• Learning profile refers to how an
individual learns best - most efficiently
and effectively.
• Teachers and their students may
differ in learning profile preferences.

26. Learning Profile Factors
Group Orientation Learning Environment
Gender
independent/self orientation quiet/noise
& warm/cool
group/peer orientation
adult orientation Culture still/mobile
combination flexible/fixed
“busy”/”spare”
Cognitive Style Intelligence Preference
analytic
Creative/conforming practical
Essence/facts creative
Expressive/controlled verbal/linguistic
Nonlinear/linear logical/mathematical
Inductive/deductive spatial/visual
People-oriented/task or Object oriented bodily/kinesthetic
Concrete/abstract musical/rhythmic
Collaboration/competition interpersonal
Interpersonal/introspective intrapersonal
Easily distracted/long Attention span naturalist
Group achievement/personal achievement existential
Oral/visual/kinesthetic
Reflective/action-oriented

27. Activity 2.5 – The Modality Preferences Instrument (HBL, p. 23)
Follow the directions below to get a score that will indicate your own modality (sense) preference(s). This instrument, keep
in mind that sensory preferences are usually evident only during prolonged and complex learning tasks.
Identifying Sensory Preferences
Directions: For each item, circle “A” if you agree that the statement describes you most of the time. Circle “D” if you
disagree that the statement describes you most of the time.
1. I Prefer reading a story rather than listening to someone tell it. A D
2. I would rather watch television than listen to the radio. A D
3. I remember faces better than names. A D
4. I like classrooms with lots of posters and pictures around the room. A D
5. The appearance of my handwriting is important to me. A D
6. I think more often in pictures. A D
7. I am distracted by visual disorder or movement. A D
8. I have difficulty remembering directions that were told to me. A D
9. I would rather watch athletic events than participate in them. A D
10. I tend to organize my thoughts by writing them down. A D
11. My facial expression is a good indicator of my emotions. A D
12. I tend to remember names better than faces. A D
13. I would enjoy taking part in dramatic events like plays. A D
14. I tend to sub vocalize and think in sounds. A D
15. I am easily distracted by sounds. A D
16. I easily forget what I read unless I talk about it. A D
17. I would rather listen to the radio than watch TV A D
18. My handwriting is not very good. A D
19. When faced with a problem , I tend to talk it through. A D
20. I express my emotions verbally. A D
21. I would rather be in a group discussion than read about a topic. A D

28. 1. I prefer talking on the phone rather than writing a letter to someone. A D
2. I would rather participate in athletic events than watch them. A D
3. I prefer going to museums where I can touch the exhibits. A D
4. My handwriting deteriorates when the space becomes smaller. A D
5. My mental pictures are usually accompanied by movement. A D
6. I like being outdoors and doing things like biking, camping, swimming, hiking etc.
A D
7. I remember best what was done rather then what was seen or talked about. A D
8. When faced with a problem, I often select the solution involving the greatest activity.
A D
9. I like to make models or other hand crafted items. A D
10. I would rather do experiments rather then read about them. A D
11. My body language is a good indicator of my emotions. A D
Interpreting the Instrument’s Score
12. Total the number of “A” responses in items 1-11
I have difficulty remembering verbal directions if I have not _____ the activity before.
done A D
This is your visual score
Total the number of “A” responses in items 12-22 _____
This is your auditory score
Total the number of “A” responses in items 23-33 _____
This is you tactile/kinesthetic score
If you scored a lot higher in any one area: This indicates that this modality is very probably your preference during a protracted and complex
learning situation.
If you scored a lot lower in any one area: This indicates that this modality is not likely to be your preference(s) in a learning situation.
If you got similar scores in all three areas: This indicates that you can learn things in almost any way they are presented.

29. Parallel Lines Cut by a Transversal
• Visual: Make posters showing all the angle
relations formed by a pair of parallel lines
cut by a transversal. Be sure to color code
definitions and angles, and state the
relationships between all possible angles.
1
2 3
4
5
6 8
7
Smith & Smarr, 2005

30. Parallel Lines Cut by a Transversal
• Auditory: Play “Shout Out!!” Given the
diagram below and commands on strips of paper
(with correct answers provided), players take turns
being the leader to read a command. The first
player to shout out a correct answer to the
command, receives a point. The next player
becomes the next leader. Possible commands:
– Name an angle supplementary 1
2 3
supplementary to angle 1. 5
8
4
6
– Name an angle congruent 7
to angle 2. Smith & Smarr, 2005

31. Parallel Lines Cut by a Transversal
• Kinesthetic: Walk It
Tape the diagram below
on the floor with masking
tape. Two players stand in 2
1
3
assigned angles. As a 5
4
team, they have to tell 6 8
what they are called (ie: 7
vertical angles) and their
relationships (ie:
congruent). Use all angle
combinations, even if
there is not a name or
relationship. (ie: 2 and 7) Smith & Smarr, 2005

32. EIGHT STYLES OF LEARNING
TYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BY
LINGUISTIC Learns through the Read Memorizing Saying, hearing and
manipulation of words. Loves names, places, seeing words
LEARNER to read and write in order to
Write
“The Word Tell stories dates and trivia
explain themselves. They also
Player” tend to enjoy talking
LOGICAL/ Looks for patterns when Do experiments Math Categorizing
solving problems. Creates a set Figure things out
Mathematical of standards and follows them
Reasoning Classifying
Learner when researching in a Work with numbers Logic Working with abstract
“The Questioner” sequential manner. Ask questions Problem solving patterns/relationships
Explore patterns and
relationships
SPATIAL Learns through pictures, charts, Draw, build, design Imagining things Visualizing
LEARNER graphs, diagrams, and art. and create things Sensing changes Dreaming
“The Visualizer” Daydream Mazes/puzzles Using the mind’s eye
Look at pictures/slides Reading maps, Working with
Watch movies charts colors/pictures
Play with machines
MUSICAL Learning is often easier for Sing, hum tunes Picking up sounds Rhythm
LEARNER these students when set to Remembering
music or rhythm
Listen to music Melody
“The Music melodies
Play an instrument Music
Lover” Noticing pitches/
Respond to music rhythms
Keeping time

33. EIGHT STYLES OF LEARNING, Cont’d
TYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BY
BODILY/ Eager to solve problems Move around Physical activities Touching
physically. Often doesn’t read (Sports/dance/ Moving
Kinesthetic directions but just starts on a
Touch and talk
Learner project Use body acting) Interacting with space
“The Mover” language crafts Processing knowledge
through bodily sensations
INTERpersonal Likes group work and Have lots of Understanding people Sharing
working cooperatively to friends Leading others Comparing
Learner solve problems. Has an
“The Socializer” interest in their community. Talk to people Organizing Relating
Join groups Communicating Cooperating
Manipulating interviewing
Mediating conflicts
INTRApersonal Enjoys the opportunity to Work alone Understanding self Working along
reflect and work Focusing inward on Individualized projects
Learner independently. Often quiet
Pursue own
“The Individual” feelings/dreams Self-paced instruction
and would rather work on his/ interests
her own than in a group. Pursuing interests/ Having own space
goals
Being original
NATURALIST Enjoys relating things to their Physically Exploring natural Doing observations
“The Nature environment. Have a strong experience nature phenomenon Recording events in Nature
Lover” connection to nature.
Do observations Seeing connections Working in pairs
Responds to Seeing patterns Doing long term projects
patterning nature Reflective Thinking

34. Introduction to Change
(MI)
• Logical/Mathematical Learners: Given a set of data that
changes, such as population for your city or town over
time, decide on several ways to present the information.
Make a chart that shows the various ways you can present
the information to the class. Discuss as a group which
representation you think is most effective. Why is it most
effective? Is the change you are representing constant or
variable? Which representation best shows this? Be ready
to share your ideas with the class.

35. Introduction to Change
(MI)
• Interpersonal
Learners: Brainstorm
things that change
constantly. Generate a list.
Discuss which of the
things change quickly and
which of them change
slowly. What would
graphs of your ideas look
like? Be ready to share
your ideas with the class.

36. Introduction to Change
(MI)
• Visual/Spatial Learners:
Given a variety of graphs, discuss
what changes each one is
representing. Are the changes
constant or variable? How can you
tell? Hypothesize how graphs
showing constant and variable
changes differ from one another.
Be ready to share your ideas with
the class.

37. Introduction to Change
(MI)
• Verbal/Linguistic Learners: Examine
articles from newspapers or magazines
about a situation that involves change
and discuss what is changing. What is
this change occurring in relation to? For
example, is this change related to time,
money, etc.? What kind of change is it:
constant or variable? Write a summary
paragraph that discusses the change and
share it with the class.

38. Multiple Intelligence Ideas for
Proofs!
• Logical Mathematical: Generate proofs for
given theorems. Be ready to explain!
• Verbal Linguistic: Write in paragraph form
why the theorems are true. Explain what
we need to think about before using the
theorem.
• Visual Spatial: Use pictures to explain the
theorem.

39. Multiple Intelligence Ideas for
Proofs!
• Musical: Create a jingle or rap to sing the
theorems!
• Kinesthetic: Use Geometer Sketchpad or
other computer software to discover the
theorems.
• Intrapersonal: Write a journal entry for
yourself explaining why the theorem is true,
how they make sense, and a tip for
remembering them.

40. Sternberg’s Three Intelligences
Creative Analytical
Practical
•We all have some of each of these intelligences, but are usually
stronger in one or two areas than in others.
•We should strive to develop as fully each of these intelligences
in students…
• …but also recognize where students’ strengths lie and teach
through those intelligences as often as possible, particularly
when introducing new ideas.

41. Thinking About the Sternberg Intelligences
ANALYTICAL Linear – Schoolhouse Smart - Sequential
Show the parts of _________ and how they work.
Explain why _______ works the way it does.
Diagram how __________ affects __________________.
Identify the key parts of _____________________.
Present a step-by-step approach to _________________.
PRACTICAL Streetsmart – Contextual – Focus on Use
Demonstrate how someone uses ________ in their life or work.
Show how we could apply _____ to solve this real life problem ____.
Based on your own experience, explain how _____ can be used.
Here’s a problem at school, ________. Using your knowledge of
______________, develop a plan to address the problem.
CREATIVE Innovator – Outside the Box – What If - Improver
Find a new way to show _____________.
Use unusual materials to explain ________________.
Use humor to show ____________________.
Explain (show) a new and better way to ____________.
Make connections between _____ and _____ to help us understand ____________.
Become a ____ and use your “new” perspectives to help us think about
____________.

42. Triarchic Theory of Intelligences
Robert Sternberg
Mark each sentence T if you like to do the activity and F if you do not like to do
the activity.
3. Analyzing characters when I’m reading or listening to a story ___
4. Designing new things ___
5. Taking things apart and fixing them ___
6. Comparing and contrasting points of view ___
7. Coming up with ideas ___
8. Learning through hands-on activities ___
9. Criticizing my own and other kids’ work ___
10. Using my imagination ___
11. Putting into practice things I learned ___
12. Thinking clearly and analytically ___
13. Thinking of alternative solutions ___
14. Working with people in teams or groups ___
15. Solving logical problems ___
16. Noticing things others often ignore ___
17. Resolving conflicts ___

43. Triarchic Theory of Intelligences
Robert Sternberg
Mark each sentence T if you like to do the activity and F if you do not like to do
the activity.
3. Evaluating my own and other’s points of view ___
4. Thinking in pictures and images ___
5. Advising friends on their problems ___
6. Explaining difficult ideas or problems to others ___
7. Supposing things were different ___
8. Convincing someone to do something ___
9. Making inferences and deriving conclusions ___
10. Drawing ___
11. Learning by interacting with others ___
12. Sorting and classifying ___
13. Inventing new words, games, approaches ___
14. Applying my knowledge ___
15. Using graphic organizers or images to organize your thoughts ___
16. Composing ___
30. Adapting to new situations ___

44. Triarchic Theory of Intelligences – Key
Robert Sternberg
Transfer your answers from the survey to the key. The column with the most
True responses is your dominant intelligence.
Analytical Creative Practical
1. ___ 2. ___ 3. ___
4. ___ 5. ___ 6. ___
7. ___ 8. ___ 9. ___
10. ___ 11. ___ 12. ___
13. ___ 14. ___ 15. ___
16. ___ 17. ___ 18. ___
19. ___ 20. ___ 21. ___
22. ___ 23. ___ 24. ___
25. ___ 26. ___ 27. ___
28. ___ 29. ___ 30. ___
Total Number of True:
Analytical ____ Creative _____ Practical _____

45. Understanding Order of Operations
Analytic Task Make a chart that shows all ways you
can think of to use order of operations
to equal 18.
Practical Task A friend is convinced that order of
operations do not matter in math. Think
of as many ways to convince your friend
that without using them, you won’t
necessarily get the correct answers!
Give lots of examples.
Creative Task Write a book of riddles that involve
order of operations. Show the solution
and pictures on the page that follows
each riddle.

46. Forms of Equations of Lines
• Analytical Intelligence: Compare and contrast the various
forms of equations of lines. Create a flow chart, a table, or
any other product to present your ideas to the class. Be
sure to consider the advantages and disadvantages of each
form.
• Practical Intelligence: Decide how and when each form of
the equation of a line should be used. When is it best to use
which? What are the strengths and weaknesses of each
form? Find a way to present your conclusions to the class.
• Creative Intelligence: Put each form of the equation of a
line on trial. Prosecutors should try to convince the jury
that a form is not needed, while the defense should defend
its usefulness. Enact your trial with group members
playing the various forms of the equations, the prosecuting
attorneys, and the defense attorneys. The rest of the class
will be the jury, and the teacher will be the judge.

47. Circle Vocabulary
All Students:
Students find definitions for a list of
vocabulary (center, radius, chord, secant,
diameter, tangent point of tangency, congruent
circles, concentric circles, inscribed and
circumscribed circles). They can use
textbooks, internet, dictionaries or any other
source to find their definitions.

48. Circle Vocabulary
Analytical
Students make a poster to explain the definitions in their own
words. Posters should include diagrams, and be easily
understood by a student in the fifth grade.
Practical
Students find examples of each definition in the room, looking
out the window, or thinking about where in the world you
would see each term. They can make a mural, picture book,
travel brochure, or any other idea to show where in the world
these terms can be seen.

49. Circle Vocabulary
Creative
Find a way to help us remember all this vocabulary!
You can create a skit by becoming each term, and
talking about who you are and how you relate to each
other, draw pictures, make a collage, or any other
way of which you can think.
OR
Role Audience Format Topic
Diameter Radius email Twice as nice
Circle Tangent poem You touch me!
Secant Chord voicemail I extend you.

50. Key Principles of a
Differentiated Classroom
• Assessment and instruction are
inseparable.
Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD

51. Pre-Assessment
• What the student already knows about what is
being planned
• What standards, objectives, concepts & skills the
individual student understands
• What further instruction and opportunities for
mastery are needed
• What requires reteaching or enhancement
• What areas of interests and feelings are in the
different areas of the study
• How to set up flexible groups: Whole, individual,
partner, or small group

52. THINKING ABOUT
ON-GOING ASSESSMENT
STUDENT DATA SOURCES TEACHER DATA
• Journal entry MECHANISMS
• Short answer test 2. Anecdotal records
• Open response test 3. Observation by checklist
• Home learning 4. Skills checklist
• Notebook 5. Class discussion
• Oral response 6. Small group interaction
• Portfolio entry 7. Teacher – student
• Exhibition conference
• Culminating product 8. Assessment stations
• Question writing 9. Exit cards
• Problem solving 10. Problem posing
11. Performance tasks and
rubrics

53. Key Principles of a
Differentiated Classroom
• The teacher adjusts content, process,
and product in response to student
readiness, interests, and learning
profile.
Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD

54. USE OF INSTRUCTIONAL
STRATEGIES.
The following findings related to
instructional strategies are supported by
the existing research:
• Techniques and instructional strategies have nearly as much influence on student
learning as student aptitude.
• Lecturing, a common teaching strategy, is an effort to quickly cover the material:
however, it often overloads and over-whelms students with data, making it likely
that they will confuse the facts presented
• Hands-on learning, especially in science, has a positive effect on student
achievement.
• Teachers who use hands-on learning strategies have students who out-perform
their peers on the National Assessment of Educational progress (NAEP) in the
areas of science and mathematics.
• Despite the research supporting hands-on activity, it is a fairly uncommon
instructional approach.
• Students have higher achievement rates when the focus of instruction is on
meaningful conceptualization, especially when it emphasizes their own knowledge
of the world.

55. Make Card Games!

56. Make Card Games!

57. Build – A – Square
• Build-a-square is based on the “Crazy” puzzles where 9
tiles are placed in a 3X3 square arrangement with all edges
matching.
• Create 9 tiles with math problems and answers along the
edges.
• The puzzle is designed so that the correct formation has all
questions and answers matched on the edges.
• Tips: Design the answers for the edges first, then write the
specific problems.
• Use more or less squares to tier. m=3
• Add distractors to outside edges and b=6 -2/3
“letter” pieces at the end.
Nanci Smith

58. The ROLE of writer, speaker, R A F T
artist, historian, etc.
An AUDIENCE of fellow writers,
students, citizens, characters, etc.
Through a FORMAT that is
written, spoken, drawn, acted, etc.
e e ron
l ct
ne r
ut on
p on
rot
A TOPIC related to curriculum
content in greater depth.

59. RAFT ACTIVITY ON FRACTIONS
Role Audience Format Topic
Fraction Whole Number Petitions To be considered Part of the
Family
Improper Fraction Mixed Numbers Reconciliation Letter Were More Alike than
Different
A Simplified Fraction A Non-Simplified Fraction Public Service A Case for Simplicity
Announcement
Greatest Common Factor Common Factor Nursery Rhyme I’m the Greatest!
Equivalent Fractions Non Equivalent Personal Ad How to Find Your Soul Mate
Least Common Factor Multiple Sets of Numbers Recipe The Smaller the Better
Like Denominators in an Unlike Denominators in an Application form To Become A Like
Additional Problem Addition Problem Denominator
A Mixed Number that 5th Grade Math Students Riddle What’s My New Name
Needs to be Renamed to
Subtract
Like Denominators in a Unlike Denominators in a Story Board How to Become a Like
Subtraction Problem Subtraction Problem Denominator
Fraction Baker Directions To Double the Recipe
Estimated Sum Fractions/Mixed Numbers Advice Column To Become Well Rounded

60. Angles Relationship RAFT
Role Audience Format Topic
One vertical angle Opposite vertical angle Poem It’s like looking in a mirror
Interior (exterior) angle Alternate interior (exterior) Invitation to a family My separated twin
angle reunion
Acute angle Missing angle Wanted poster Wanted: My complement
An angle less than 180 Supplementary Persuasive speech Together, we’re a straight angle
angle
**Angles Humans Video See, we’re everywhere!
** This last entry would take more time than the previous 4 lines, and assesses a little differently. You could offer it as
an option with a later due date, but you would need to specify that they need to explain what the angles are, and anything
specific that you want to know such as what is the angle’s complement or is there a vertical angle that corresponds, etc.

61. Algebra RAFT
Role Audience Format Topic
Coefficient Variable Email We belong together
Scale / Balance Students Advice column Keep me in mind
when solving an
equation
Variable Humans Monologue All that I can be
Variable Algebra students Instruction manual How and why to
isolate me
Algebra Public Passionate plea Why you really do
need me!

62. RAFT Planning Sheet
Know
Understand
Do
How to Differentiate:
• Tiered? (See Equalizer)
• Profile? (Differentiate Format)
• Interest? (Keep options equivalent in
learning)
• Other?
Role Audience Format Topic

63. Cubing
Cubing
Ideas for Cubing Cubing
• Arrange ________ into a 3-D collage
to show ________ Ideas for Cubing in Math
• Make a body sculpture to show • Describe how you would solve ______
________ • Analyze how this problem helps us use
mathematical thinking and problem solving
• Create a dance to show • Compare and contrast this problem to one
• Do a mime to help us understand on page _____.
• Present an interior monologue with • Demonstrate how a professional (or just a
dramatic movement that ________ regular person) could apply this kink or
problem to their work or life.
• Build/construct a representation of • Change one or more numbers, elements, or
________ signs in the problem. Give a rule for what
• Make a living mobile that shows and that change does.
balances the elements of ________ • Create an interesting and challenging word
problem from the number problem. (Show us
• Create authentic sound effects to how to solve it too.)
accompany a reading of _______ • Diagram or illustrate the solutionj to the
• Show the principle of ________ with a problem. Interpret the visual so we
understand it.
rhythm pattern you create. Explain to
us how that works.

64. Describe how you would Explain the difference
1 3
solve + or roll between adding and
5 5
the die to determine your multiplying fractions,
own fractions.
Compare and contrast Create a word problem
these two problems: that can be solved by
1 2 11
+ =
+ 3 5 15
and (Or roll the fraction die to
1 1
+ determine your fractions.)
3 2
Describe how people use Model the problem
fractions every day. ___ + ___ .
Nanci Smith
Roll the fraction die to
determine which fractions
to add.

65. Nanci Smith

66. Describe how you would Explain why you need
2 3 1
solve + + or roll a common denominator
13 7 91
the die to determine your when adding fractions,
own fractions. But not when multiplying.
Can common denominators
Compare and contrast ever be used when dividing
these two problems: fractions?
1 1 3 1
+ and +
3 2 7 7
Create an interesting and
challenging word problem
A carpet-layer has 2 yards that can be solved by
of carpet. He needs 4 feet ___ + ____ - ____.
of carpet. What fraction of Roll the fraction die to
his carpet will he use? How determine your fractions.
Nanci Smith do you know you are correct?
Diagram and explain the
solution to ___ + ___ + ___.
Roll the fraction die to
determine your fractions.

67. Level 1:
1. a, b, c and d each represent a different value. If a = 2, find
b, c, and d.
a+b=c
a–c=d
a+b=5
2. Explain the mathematical reasoning involved in solving
card 1.
3. Explain in words what the equation 2x + 4 = 10 means.
Solve the problem.
4. Create an interesting word problem that is modeled by
8x – 2 = 7x.
5. Diagram how to solve 2x = 8.
6. Explain what changing the “3” in 3x = 9 to a “2” does to
the value of x. Why is this true?

68. Level 2:
1. a, b, c and d each represent a different value. If a = -1, find
b, c, and d.
a+b=c
b+b=d
c – a = -a
2. Explain the mathematical reasoning involved in solving
card 1.
3. Explain how a variable is used to solve word problems.
4. Create an interesting word problem that is modeled by
2x + 4 = 4x – 10. Solve the problem.
5. Diagram how to solve 3x + 1 = 10.
6. Explain why x = 4 in 2x = 8, but x = 16 in ½ x = 8. Why
does this make sense?

69. Level 3:
1. a, b, c and d each represent a different value. If a = 4, find b,
c, and d.
a+c=b
b-a=c
cd = -d
d+d=a
2. Explain the mathematical reasoning involved in solving
card 1.
3. Explain the role of a variable in mathematics. Give examples.
4. Create an interesting word problem that is modeled by
3x − 1 ≤ 5 x + 7. Solve the problem.
5. Diagram how to solve 3x + 4 = x + 12.
6. Given ax = 15, explain how x is changed if a is large or a is
small in value.

70. Designing a Differentiated Learning
Contracthas the following components
A Learning Contract
2. A Skills Component
Focus is on skills-based tasks
Assignments are based on pre-assessment of students’ readiness
Students work at their own level and pace
3. A content component
Focus is on applying, extending, or enriching key content (ideas, understandings)
Requires sense making and production
Assignment is based on readiness or interest
4. A Time Line
Teacher sets completion date and check-in requirements
Students select order of work (except for required meetings and homework)
4. The Agreement
The teacher agrees to let students have freedom to plan their time
Students agree to use the time responsibly
Guidelines for working are spelled out
Consequences for ineffective use of freedom are delineated
Signatures of the teacher, student and parent (if appropriate) are placed on the agreement
Differentiating Instruction: Facilitator’s Guide, ASCD, 1997

71. Montgomery
Personal Agenda County, MD
Personal Agenda for _______________________________________
Starting Date _____________________________________________________
Teacher & student
initials at Special
Task
completion Instructions
Remember to complete your daily planning log; I’ll call on you for
conferences & instructions.

72. Proportional Reasoning
Think-Tac-Toe
□ Create a word problem that □ Find a word problem from □ Think of a way that you use
requires proportional the text that requires proportional reasoning in your
reasoning. Solve the proportional reasoning. life. Describe the situation,
problem and explain why it Solve the problem and explain why it is proportional
requires proportional explain why it was and how you use it.
reasoning. proportional.
□ Create a story about a □ How do you recognize a □ Make a list of all the
proportion in the world. proportional situation? proportional situations in the
You can write it, act it, Find a way to think about world today.
video tape it, or another and explain proportionality.
story form.
□ Create a pict-o-gram, poem □ Write a list of steps for □ Write a list of questions to ask
or anagram of how to solve solving any proportional yourself, from encountering a
proportional problems problem. problem that may be
proportional through solving
it.
Directions: Choose one option in each row to complete. Check the box of the choice you make, and turn
this page in with your finished selections.
Nanci Smith, 2004

73. Similar Figures Menu
Imperatives (Do all 3):
2. Write a mathematical definition of “Similar Figures.” It
must include all pertinent vocabulary, address all
concepts and be written so that a fifth grade student
would be able to understand it. Diagrams can be used to
illustrate your definition.
3. Generate a list of applications for similar figures, and
similarity in general. Be sure to think beyond “find a
missing side…”
4. Develop a lesson to teach third grade students who are
just beginning to think about similarity.

74. Similar Figures Menu
Negotiables (Choose 1):
2. Create a book of similar figure applications and
problems. This must include at least 10 problems. They
can be problems you have made up or found in books,
but at least 3 must be application problems. Solver each
of the problems and include an explanation as to why
your solution is correct.
3. Show at least 5 different application of similar figures in
the real world, and make them into math problems.
Solve each of the problems and explain the role of
similarity. Justify why the solutions are correct.

75. Similar Figures Menu
Optionals:
2. Create an art project based on similarity. Write a cover
sheet describing the use of similarity and how it affects
the quality of the art.
3. Make a photo album showing the use of similar figures
in the world around us. Use captions to explain the
similarity in each picture.
4. Write a story about similar figures in a world without
similarity.
5. Write a song about the beauty and mathematics of
similar figures.
6. Create a “how-to” or book about finding and creating
similar figures.

76. Whatever it Takes!